i will present joint work with jacek jendrej (crns, sorbonne paris nord) on equivariant wave maps with values in the two-sphere. we prove that every finite energy equivariant wave map resolves, as time passes, into a superposition of decoupled harmonic maps and radiation, settling the soliton resolution conjecture for this equation.  it was proved in works of côte, and jia and kenig, that such a decomposition holds along a sequence of times. we show the resolution holds continuously-in-time via a “no-return” lemma based on the virial identity. the proof combines a collision analysis of solutions near a multi-soliton configuration with concentration compactness techniques. as a byproduct of our analysis we also prove that there are no elastic collisions between pure multi-solitons.